True or False: In any situation that has only two possible outcomes, each outcome will have probability 1/2?

True.

Explanation:

In situations with only two possible outcomes, it is often assumed that each outcome has an equal chance of occurring, resulting in a probability of 1/2 or 50% for each outcome. This assumption is based on the principle of symmetry and is known as the principle of indifference or the principle of equally likely outcomes.

Principle of Equally Likely Outcomes:

The principle of equally likely outcomes states that if all outcomes in a sample space are equally likely, then the probability of any particular outcome occurring is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

For example, consider flipping a fair coin. The sample space consists of two outcomes: heads (H) or tails (T). Since the coin is fair, it is assumed that both outcomes are equally likely. Therefore, the probability of obtaining heads is 1/2, and the probability of obtaining tails is also 1/2.

Applications:

The assumption of equal probabilities for two outcomes is often applied in various scenarios, such as:

1. Coin Toss: As mentioned earlier, when flipping a fair coin, the probability of getting heads or tails is considered to be 1/2 each.

2. Dice Roll: When rolling a fair six-sided die, each face has an equal chance of landing up, resulting in a probability of 1/6 for each outcome.

3. Binary Events: Many real-life situations can be modeled as binary events, where there are only two possible outcomes. For instance, a binary event could be the success or failure of a particular task, and if the outcomes are assumed to be equally likely, each outcome would have a probability of 1/2.

Limitations:

While assuming equal probabilities for two outcomes is often useful and convenient in many situations, it may not always reflect the true probabilities in complex scenarios. Factors such as bias, uneven distribution, or different underlying probabilities can affect the actual likelihood of each outcome.

Conclusion:

In situations with only two possible outcomes, it is generally assumed that each outcome has a probability of 1/2. However, it is essential to recognize that this assumption may not always hold true in every scenario, and careful consideration must be given to the specific context and underlying probabilities involved.